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Mechanics of Materials Chapter 2 Strain

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Mechanics of Materials Chapter 2 Strain Mechanics of Materials Chapter 2 Strain 2.1 Introduction So far dealt mainly with the strength of structural member. Here we being our study of an equally important topic of mechanics －deformations, or strains. In general terms, strain is a geometric quantity that measures the deformation of a body. There are two types of strain：normal strain, which characterizes dimensional changes, and shear strain, which describes distortion (changes in angles). Stress and strain are two fundamental concepts of mechanics of materials. Their relationship to each other defines the mechanical properties of a material, the knowledge of which is of the utmost importance in design. Use force-deformation relationships in conjunction with equilibrium analysis to solve statically indeterminate problems. 2.2 Axial Deformation；Stress-Strain Diagram The strength of a material is not the only criterion that must be considered when designing machine parts or structures. The stiffness of a material is often equally important properties such as hardness, toughness, and ductility.These properties are determined by laboratory tests. Many materials, particularly metals, have established standards that describe the test procedures in detail. We will confine our attention to only one of the tests－the tensile test of steel－ and use its results to illustrate several important concepts of material behavior. a. Normal (axial) strain The elongation δmay be caused by an applied axial force, or an expansion due to an increase in temperature, or even a force and a temperature increase acting simultaneously. Figure 2.1 Deformation of a prismatic bar. Strain describes the geometry of deformation. The normal strain ε(lowercase Greek epsilon) is defined as the elongation per unit length. Therefore, the normal strain in the bar in the axial direction, also known as the axial strain, is δ ε = (2.1) L If the bar deforms uniformly, then Eq. (2.1) expression should be viewed as the average axial strain. Note that normal strain, being elongation per unit length, is a dimensionless quantity. However, “units”such as in./in. or mm/mm are frequently used for normal strain. If the deformation is not uniform, we let O be a point in the bar located at the distance χfrom the fixed end. We define the axial strain at point O as Δδ dδ ε = lim = (2.2) Δx → o Δχ dχ We note that if the distribution of the axial strain ε is known, the elongation of the can be computed from L L δ = ∫∫dδ = εdχ O O (2.3) For uniform strain distribution, Eq. (2.3) yields δ= ε L, which agrees with Eq.(2.1). The results are also applicable to compression. By convention, compression (shortening) carries a negative sign. b. Tension test In the standard tension test, the specimen shown in Fig.2.2 is placed in the grips of a testing machine. The grips are designed so that the load P applied by the machine is axial. Two gage marks are scribed on the specimen to define the gage length L. These marks are loaded away from the ends to avoid the load effects caused by the grips and to ensure that the stress and strain are uniform in the material between the marks. The testing machine elongates the specimen at a slow, constant rate until the specimen ruptures. ( )g Figure 2.2 Specimen used in the standard tension test. During the test, continuous readings are taken of the applied load and the elongation of the gage length.These data are then converted to stress and strain. The stress is obtained from σ=P/A, where P is the load and A represents the original cross-sectional area of the specimen. The strain is computed from ε =σ/L, where δis the elongation where δis the elongation Between the gage marks and L is the original gage length. There results are referred to as nominal stress and nominal strain. As the bar is being stretched, its cross-sectional area is reduced and the length between the gage marks increases. Dividing the load by the actual (current) area of the specimen, we get the true stress. Similarly, the true strain is obtained by dividing the elongation δby the current gage length. The nominal and true measures are essentially the same in the working range of metals. They differ only for very large strains, With only a few exceptions, engineering applications use nominal stress and strain. ( )gg A stress-strain diagram for structural steel is shown in Fig. 2.3. The following mechanical properties can be determined from the diagram. Figure 2.3 Stress-strain diagram obtained from the standard tension test on a structural steel specimen. Proportional Limit and Hooke’s Law As seen in Fig. 2.3, the stress-strain diagram is a straight line from the origin O to a point called the proportional limit. This plot is a manifestation of Hooke’s law : Stress is proportional to strain; that is, σ= E Є (2.4) where E is material property known as the modulus of elasticity or Young’s modulus. The units of E are the same as the units of, Pa or psi. For steel, E =29×106 psi, or 200 GPa, approximately. ( )gg Note that Hooke’s law does not apply to the entire diagram；its validity ends at the proportional limit. Beyond this point, stress is no longer proportional to strain. Elastic Limit A material is said to be elastic if, after being loaded, the material returns to its original shape when the load is removed. The stress beyond which the material is no longer elastic. The permanent deformation that remains after the removal of the load is called the permanent set. The elastic limit is slightly larger than the proportional limit. Yield Point The point where the stress-strain diagram becomes almost horizontal is called the yield point, and the corresponding stress is known as the yield stress or yield strength. Beyond the yield ( )gg point there is an appreciable elongation of the material without a corresponding increase in load. Indeed, the load may actually decrease while the yielding occurs. However, the phenomenon of yielding is unique to structural steel. Other grades of steel, steel alloys, and other material do not yield, as indicated by the stress-strain curves of the materials shown in Fig.2.4.. After repeated loading, these residual stresses are removed and the stress- ( )2003 k /Cl blihi / h i strain curves become practically straight lines. Figure 2.4 Stress-strain diagrams for various materials that fail without significant yielding. For materials that do not have ( )gg a well-defined yield point, yield stress is determined by the offset method. This method consists of drawing a line parallel to the initial tangent of the stress-strain curve；this line starts at a prescribed offset strain, usually 0.2 % ( ε = 0.002 ). The intersection of this line with the stress-strain curve, is Figure 2.5 Determining the yield called the yield point at 0.2 point by the 2% offset method. % offset. Ultimate Stress The ultimate stress or ultimate strength, as it is often called, is the highest stress on the stress-strain curve. Rupture Stress The rupture strength is the stress at which failure occurs. The nominal rupture strength is computed by dividing the load at rupture by the original cross-sectional area. The true rupture strength is calculated using the reduced area of the cross section where the fracture occurred. The difference in the two values results from a phenomenon known as necking. As failure approaches, the material stretches very rapidly, causing the cross section to narrow, as shown in Fig. 2.6. However, the ultimate strength is commonly used as the maximum stress that the material can carry. Figure 2.6 Failed tensile test specimen howing necking, or narrowing, of the cross section. C. Working stress and factor of safety The working stress σw, also called the allowable stress, is the maximum safe axial stress used in design. In most design, the working stress should be limited to values not exceeding the proportional limit so that the stresses remain in the elastic range. However, because the proportional limit is difficult to determine accurately, it is customary to base the working stress on either the yield stress σyp or the ultimate stress σult, divided by a suitable number N, called the factor of safety. Thus, σ σ σ = yp or ult (2.5) w N N The yield point is selected as the basis for determining σw in structural steel because it is the stress at which a prohibitively large permanent set may occur. For other material, the working stress is usually based on the ultimate strength. usually the working stress is set by a group of experienced engineers and is embodied in building codes and specifications. In many materials the proportional limit is about one-half the ultimate strength. To avoid accidental overloading, a working stress of one- half the proportional limit is usually specified for dead loads that are gradually applied (The term dead load refers to the weight of the structure and other loads that, once applied are not removed.) A working stress set in this way corresponds to a factor of safety of 4 with respect to σult and is recommended for materials that are known to be uniform and homogenous. For other materials, such as wood, in which unpredictable nonuniformities ( such as knotholes) may occur, larger factors of safety are used. The dynamic effect of suddenly applied loads also requires higher factors of safety. 2.3 Axially Loaded Bars The stress caused by P is below the proportional limit, so that Hooke`s law σ= E·ε is applicable.
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